Prediction of chatter stability for milling process using Runge-Kutta-based complete discretization method

On the basis of the classical Runge-Kutta method and the complete discretization method, a Runge-Kutta-based complete discretization method (RKCDM) is proposed in the paper to predict the chatter stability of milling process, in which the regenerative effect is taken into consideration. Firstly, the dynamics model of milling process is simplified as a 2-DOF vibration system in the two orthogonal directions, which can be expressed as coefficient-varying periodic differential equations with a single time delay. Then, all parts of the delay differential equation (DDE), including delay term, time-domain term, parameter matrices, and most of all the differential terms are discretized using the classical fourth-order Runge-Kutta iteration method to replace the direct integration scheme used in the classical semi-discretization method (C-SDM) and the classical complete discretization scheme with the Euler method (C-CDSEM), which can simplify the complexity of the discretization iteration formula greatly. Lastly, the Floquet theory is adopted to predict the stability of milling process by judging the eigenvalues of the state transition matrix corresponding to certain cutting conditions. Comparing RKCDM with C-SDM and C-CDSEM, the numerical simulation results show that RKCDM has the highest convergence rate, computation accuracy, and computation efficiency. As dichotomy search rather than sequential search is used in the algorithm, the calculation time for obtaining the stability lobe diagrams (SLDs) is greatly reduced. As a result, it is practical to determine the optimal chatter-free cutting conditions for milling operation in shop floor applications.